LaTeX
\bar x\pm z_{\alpha/2}\frac{\sigma}{\sqrt n}Variables
- x̄: sample mean
- z: critical value
- σ: population standard deviation
- n: sample size
How to use this formula
Estimates a population mean when population standard deviation is known.
Important notes
- Use a t critical value when σ is unknown and assumptions apply.
Quick example
For 95%, z≈1.96.
Applicability, worked calculation, and verification
Assumptions and domain checks
- For the Mean Confidence Interval, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.
- For a real-valued result, each even-root radicand must be nonnegative unless complex values are intended.
- The plus-minus sign represents separate branches; evaluate and verify both when both are admissible.
- For the Mean Confidence Interval, identify whether each quantity is a sample statistic, population parameter, estimator, or model value, and check the method assumptions.
Worked example
For 95%, z≈1.96.
Common mistakes
- Do not collapse the ± in Mean Confidence Interval into one value; calculate each admissible branch and check it in the original problem.
- For the Mean Confidence Interval, do not interpret a descriptive statistic as a causal or population conclusion without the sampling and model assumptions.
Continue the workflow
Use Mean Confidence Interval in your own work
- Check the domainMatch the variables and assumptions to the problem before substituting values.
- Copy the exact notationPreserve grouping, signs, and exponents in
\bar x\pm z_{\alpha/2}\frac{\sigma}{\sqrt n}. - Edit or convertOpen the expression in the LaTeX editor, then export it for your document or web page.
Review and verification
Last reviewed: 2026-07-23
Automated quality check: Kept noindex until the missing evidence is supplied.
Formula references
- Introductory Statistics 2eOpenStax, Rice University — Reviewed probability and statistics definitions, notation, and formulas.
Frequently asked questions
What is the Mean Confidence Interval used for?
Estimates a population mean when population standard deviation is known.
Can I copy this formula as LaTeX?
Yes. Copy \bar x\pm z_{\alpha/2}\frac{\sigma}{\sqrt n} or open it in the LaTeX editor.
What should I check before using it?
Confirm that each variable, unit, domain restriction, and assumption matches the problem.